The first four Laguerre polynomials are 1, 1- 1,2-4t +t2, and 6 18t + 9t2-1. Show that these polynomials form a basis of P3.
The first four Laguerre polynomials are 1, 1- 1,2-4t +t2, and 6 18t + 9t2-1. Show that these polynomials form a basis of P3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Number 22 linear algebra
![V and if T is a set of more than p vectors in V,
mensional vector
then T is linearly dependent.
20. a. R2 is a two-dimensional subspace of R'.
b. The number of variables in the equation Ax =0 equals
the dimension of Nul A.
= = %}
c. A vector space is infinite-dimensional if it is spanned by
an infinite set.
d. If dim V =n and if S spans V, then S is a basis of V.
s in R' whose
e. The only three-dimensional subspace of R' is R' itself.
21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and
-12t + 813. These polynomials arise naturally in the study
of certain important differential equations in mathematical
physics.? Show that the first four Hermite polynomials form
a basis of P3.
2
spanned by
the subspace
22. The first four Laguerre polynomials are 1, 1-1,2-4t +1,
and 6 18t +9t2-t. Show that these polynomials form a
basis of P3.
23. Let B be the basis of P, consisting of the Hermite polynomi-
als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find the
coordinate vector of p relative to B.
24. Let B be the basis of P, consisting of the first three
Laguerre polynomials listed in Exercise 22, and let
p(t) = 5+5t - 2t2. Find the coordinate vector of p relative
For the matrices
to B.
25. Let S be a subset of an n-dimensional vector space V, and
suppose S contains fewer than n vectors. Explain why S
cannot span V.
26. Let H be an n-dimensional subspace of an n-dimensional
vector space V. Show that H =V.
27. Explain why the space P of all polynomials is an infinite-
dimensional space.
2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor and
David C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Other
sets of polynomials are discussed there, too.
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F6a5813e6-dea4-4aea-ad89-09312a299012%2Fjt7jbnr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:V and if T is a set of more than p vectors in V,
mensional vector
then T is linearly dependent.
20. a. R2 is a two-dimensional subspace of R'.
b. The number of variables in the equation Ax =0 equals
the dimension of Nul A.
= = %}
c. A vector space is infinite-dimensional if it is spanned by
an infinite set.
d. If dim V =n and if S spans V, then S is a basis of V.
s in R' whose
e. The only three-dimensional subspace of R' is R' itself.
21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and
-12t + 813. These polynomials arise naturally in the study
of certain important differential equations in mathematical
physics.? Show that the first four Hermite polynomials form
a basis of P3.
2
spanned by
the subspace
22. The first four Laguerre polynomials are 1, 1-1,2-4t +1,
and 6 18t +9t2-t. Show that these polynomials form a
basis of P3.
23. Let B be the basis of P, consisting of the Hermite polynomi-
als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find the
coordinate vector of p relative to B.
24. Let B be the basis of P, consisting of the first three
Laguerre polynomials listed in Exercise 22, and let
p(t) = 5+5t - 2t2. Find the coordinate vector of p relative
For the matrices
to B.
25. Let S be a subset of an n-dimensional vector space V, and
suppose S contains fewer than n vectors. Explain why S
cannot span V.
26. Let H be an n-dimensional subspace of an n-dimensional
vector space V. Show that H =V.
27. Explain why the space P of all polynomials is an infinite-
dimensional space.
2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor and
David C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Other
sets of polynomials are discussed there, too.
3.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)